Abstract

We provide further insight into why the inverse rule [J. Opt. Soc. Am. A 13, 1870 (1996)] for multiplying two finite Fourier series of two pairwise discontinuous functions yields correct results at the point of discontinuity.

© 2000 Optical Society of America

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References

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  1. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  3. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  4. E. Butkov, Mathematical Physics (Addison-Wesley, New York, 1968).

1996 (2)

1983 (1)

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Equations (9)

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f(x)=a, -Λ/2<x<0b, 0<x<Λ/2,
g(x)=b, -Λ/2<x<0a, 0<x<Λ/2.
hT(2)(0)=11a+1b2a+b2=ab=h(0),
11a+1b2.
hT(1)(x)=fT(x)gT(x)=[c0+DN(x)][c0-DN(x)],
DN(x)=n=1Ncn sin nKx,c0=(b+a)/2,
cn=(2/πn)(b-a),nodd,cn=0,neven,
hT(2)(x)=(1/fTREC)gT(x)=1(1/ab)gTgT=ab,
forall x.

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